Distributions of Finite Sequences Represented by Polynomials in Piatetski-Shapiro Sequences

TL;DR

This paper investigates the distribution of finite sequences generated by Piatetski-Shapiro sequences, showing their representation by polynomials in rational coefficients and analyzing the asymptotic density of such sequences.

Contribution

It establishes the asymptotic density of sequences represented by polynomials within Piatetski-Shapiro sequences, extending results to Hardy field functions and considering fixed and variable common differences.

Findings

Asymptotic density for d=1 is 1/(k-1)

Sequences can be represented by polynomials of degree at most d

Results extend to functions in Hardy fields

Abstract

By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $\alpha\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence $(\lfloor{(n+rj)^\alpha}\rfloor)_{j=0}^{k-1}$ is represented as $\lfloor{(n+rj)^\alpha}\rfloor=p(j)$, $j=0,1,\ldots,k-1$, by using some polynomial $p(x)\in\mathbb{Q}[x]$ of degree at most $d$. In particular, the above sequence is an arithmetic progression when $d=1$. In this paper, we show the asymptotic density of such numbers $n$ as above. When $d=1$, the asymptotic density is equal to $1/(k-1)$. Although the common difference $r$ is arbitrarily fixed in the above result, we also examine the case when $r$ is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields.